Let \(([0,1],\Sigma,\mu)\) be a nonatomic measure space and let \(\mathcal H\) be a separable complex Hilbert space with norm \({\|\cdot\|_2}\). Let \(p(t)\) be a measurable function from \([0,1]\) into \((1,\infty)\) such that \(1<p_0\leq p(t)\leq p_\infty<\infty\). The authors consider the Nakano space \(N=L^{p(t)}(\mu, {\mathcal H} )\). For any \(f\in N\), let \[ M(f)=\int_0^1\frac{\|f(t)\|_2^{p(t)}}{p(t)}\;d\mu(t), \]
and the norm of \(f\) in \(N\) is \(\|f\|_N=\inf\{\varepsilon>0: M(f/\varepsilon)\leq 1\}\). On this space \(N\), the authors are interested in the form of the Hermitian operators and the form of the surjective isometries. The following theorem gives a characterization of the Hermitian operators on \(N\).
Theorem. The operator \(H\) is a Hermitian operator on \(N\) if and only if there is a strongly measurable map \(A\) of \([0,1]\) such that \(A(\cdot)\) is a Hermitian \(B(\mathcal H)\)-valued function, \(A(\cdot){\mathbf z}\in N\), \(\|A(t)\|\leq \|H\|\) and for every \(f\) in \(N\)
\[ (Hf)(t)=A(t)f(t) \;\;\text{ almost everywhere.} \]
For arbitrary vector \(z\in \mathcal H\), we denote by \(\mathbf z\) the constant function \(z(t)=z\) for every \(t\in [0,1]\).
The following part of Theorem 13 gives a representation of the surjective isometries on \(N\).
Theorem. If \(U\) is a surjective isometry on \(N\), then there is a regular set isomorphism \(\varphi^{-1}\) of \(\Sigma\), a strongly measurable map \(V\) of \([0,1]\) into \(B(\mathcal H)\) such that \(V(t)\) is an isometry of \(\mathcal H\) onto itself for almost all \(t\in[0,1]\), and \(u\) is a measurable function that satisfies \(\mu(\sigma)=\int_{\varphi(\sigma)} u(t)d\mu(t)\) such that \(p(t)=p(\varphi^{-1}(t))\) almost everywhere and
\[ Uf(t)=[u(\varphi(t))]^{-1/p(t)}V(\varphi(t))f(\varphi(t)). \]